Cosmological Correlators Using Tensor Networks

Abstract

We develop a nonperturbative tensor-network framework for computing cosmological correlators in de Sitter space and use it to test the proposal that suitably defined in-in correlators can be obtained from an in-out formalism by gluing the expanding and contracting Poincaré patches. Focusing on interacting $1+1$-dimensional $φ^4$ theory, we formulate finite-time lattice observables using Matrix Product State (MPS) techniques and analyze the regulator subtleties associated with the singular behavior near the patching surface. Within this regulated framework, we find controlled nonperturbative evidence for the proposed relation between in-in and in-out correlators in several examples. We also find suggestive evidence that the perturbative obstructions present for sufficiently light fields can be softened nonperturbatively, albeit in a regime of substantially larger entanglement. A central outcome of our analysis is an entanglement-based picture of the computation: for in-in evolution the entanglement remains modest and can decrease toward late times, whereas in the patched in-out set-up it grows significantly after the gluing slice. Thus, although the in-out formalism is perturbatively economical, the in-in formulation is numerically more favorable. We briefly discuss how the same strategy extends to low-angular-momentum sectors in $3+1$ dimensions, and why regimes of rapid entanglement growth may eventually motivate quantum-computing implementations.

Figures (11)

• Figure 1: Schematics of in-in and in-out formalisms. In the in-in formalism, only one copy of dS$_2$ is needed. The state is prepared at $\eta_i$ and operators are inserted at $\eta_*$ (marked by the red crosses). Evolutions are denoted by the arrows. The in-in correlator is independent of the reference time $\eta_{\rm ref}$ as long as $\eta_{\rm ref}>\eta_*$. In the in-out formalism, we need to glue two copies of dS$_2$ along $\eta=0$. The in and out states are at $\eta_i$ and $\eta_f$ respectively.
• Figure 2: Interaction effects on non-Gaussianity and mode production. The left panel shows the dimensionless excess kurtosis $\kappa_{\rm rel}^{\rm int}(\eta)$ defined in \ref{['eq:kappa_def']} for $\lambda=0.1,0.5,1$. Increasing $\lambda$ drives a larger departure from Gaussianity, and $\kappa_{\rm rel}^{\rm int}(\eta)$ becomes negative at late times, indicating a platykurtic distribution. The right panel shows the cumulative interaction-induced shift $S_{\rm int}(J;\lambda)$ defined in \ref{['eq:Sint_def']}, obtained from the production diagnostic $\Delta N_j^{\rm prod}$ built from the instantaneous mode occupation \ref{['eq:Nj_def']}. The negative plateau indicates suppression of production relative to the free $(\lambda=0)$ squeezing baseline, while the rapid saturation shows that this effect is dominated by the infrared modes, with $J\lesssim 10$ already capturing most of it in this example.
• Figure 3: Top panel: Convergence of in--in (circles) to the real part of in--out (square) values for different $N_{\rm max}$ as a function of the bond dimension $\chi$. Bottom panel: dependence of $\mathfrak{I}\{B_{\rm in-out}\}$ on $\chi$. We also plotted the perturbative benchmarks. For this simulation, Approach B is used and $N=11$, $m^2=1$, $\eta_i=-10$, $\eta_*=-1$, $\eta_f=10$, $\eta_0=0.3$.
• Figure 4: Real and imaginary parts for $C(r, \eta_*)$ with $\chi = 30$ (top) and $\chi = 70$ (bottom), $N = 40, N_{\rm max} = 15, \eta_* = -1, \eta_i = -20, \eta_f = 20, m^2 = 1, \lambda = 0.1, \eta_0 = 0.3, \mu = 0.3$. Filled markers denote positive values whereas hollow ones denote negative. Increasing the bond dimension improves the agreement at larger values of $r$. However, close to zero crossing there is a large disagreement in the values of the in--in and in--out correlators. Approach A was used.
• Figure 5: Midpoint correlator $\langle\phi^2\rangle$ over the $m^2$--$\lambda$ parameter space with $N=11$, $\chi=30$, $N_{\rm max}=12$, $\eta_*=-2.5$, $\eta_0=0.3$. Left: Relative difference between the real parts of the in-in and in-out correlators, $|\Re\langle\phi^2\rangle_{\mathrm{in\text{-}in}} - \Re\langle\phi^2\rangle_{\mathrm{in\text{-}out}}| / |\Re\langle\phi^2\rangle_{\mathrm{in\text{-}in}}|$. Right: Imaginary part of the in-out correlator, $\Im\langle\phi^2\rangle_{\mathrm{in\text{-}out}}$.